WEAKLY COUPLED HAMILTON-JACOBI SYSTEMS WITHOUT MONOTONICITY CONDITION: A FIRST STEP

In this paper, we mainly focus on the existence of the viscosity solutions of {H-1(x, Du(1)(x), u(1)(x), u(2)(x)) = 0, H-2(x, Du(2)(x), u(2)(x), u(1)(x)) = 0. The standard assumption for the above system is called the monotonicity condition, which requires that H-i is increasing in u(i) and decreasing in u(j) for each i, j is an element of {1, 2} and i not equal j. In this paper, it is assumed that H-i is either increasing or decreasing in u(i), and may be non-monotone in u(j). The existence of viscosity solutions is proved when chi := sup(u,v,w is an element of R) vertical bar partial derivative u(2)H(1)(x, 0, 0, u)/partial derivative H-u1(1)(x, 0, v, w)vertical bar center dot sup(u,v,omega is an element of R)vertical bar partial derivative H-u1(2)(x, 0, 0, u)/partial derivative H-u2(2)(x, 0, v, w)vertical bar < 1. Then we consider {h(1)(x, Du(1)(x)) + Lambda(1)(x)(u(1)(x) - u(2)(x)) = c, h(2)(x, Du(2)(x)) + Lambda(2)(x)(u(2)(x) - u(1)(x)) = alpha(c). It turns out that for each c is an element of R, there is a unique constant alpha(c) is an element of R such that the above system has viscosity solutions. The function c bar right arrow alpha(c) is non-increasing and Lipschitz continuous. In the appendix, the large time convergence of the viscosity solution of evolutionary weakly coupled systems is proved when chi < 1.